SOLUTION: if the product of two consecutive integers is decreased by 20 times the greater integer, the result is 442.find the integers.

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Question 168019This question is from textbook
: if the product of two consecutive integers is decreased by 20 times the greater integer, the result is 442.find the integers.This question is from textbook

Found 2 solutions by jojo14344, KnightOwlTutor:
Answer by jojo14344(1512) About Me  (Show Source):
You can put this solution on YOUR website!
Let consecutive inetgers:
x= 1st integer
x%2B1=2nd integer
Condition:
%28x%29%28x%2B1%29-20%28x%2B1%29=442, right? ------------------> Eqn 1
Continuing,
x%5E2%2Bx-20x-20-442=0
x%5E2-19x-462=0
where----system%28a=1%2Cb=-19%2Cc=-462%29
By Pyth. Theorem:
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
x=%28-%28-19%29%2B-sqrt%28-19%5E2-4%2A1%2A-462%29%29%2F%282%2A1%29
x=%2819%2B-sqrt%28361%2B1848%29%29%2F2
x=%2819%2B-sqrt%282209%29%29%2F2
x=%2819%2B-47%29%2F2
2 values: x=%2819%2B47%29%2F2=66%2F2=highlight%2833%29
x=%2819-47%29%2F2=-28%2F2=-14
Use highlighted one=33, 1st integer; 34 = 2nd inetger (Answer)
Go back Eqn 1 to check:
33%2833%2B1%29-20%2833%2B1%29=442
33%2A34-20%2A34=442
1122-680=442
442=442
Thank you,
Jojo

Answer by KnightOwlTutor(293) About Me  (Show Source):
You can put this solution on YOUR website!
X=first #
x+1=2nd #
The equation is as follows
(x)(x+1)-20(x+1)=442
mulitply the terms to get:
x^2+x-20x-20=442
Combine like terms
x^2-19x-20=442
Add 20 to both sides
x^2-19x=462
Subtract 462 from both sides
x^2-19x-462=0
This is a quadratic eqation or parabola
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-19x%2B-462+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-19%29%5E2-4%2A1%2A-462=2209.

Discriminant d=2209 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--19%2B-sqrt%28+2209+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-19%29%2Bsqrt%28+2209+%29%29%2F2%5C1+=+33
x%5B2%5D+=+%28-%28-19%29-sqrt%28+2209+%29%29%2F2%5C1+=+-14

Quadratic expression 1x%5E2%2B-19x%2B-462 can be factored:
1x%5E2%2B-19x%2B-462+=+1%28x-33%29%2A%28x--14%29
Again, the answer is: 33, -14. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-19%2Ax%2B-462+%29


The solutions to this equation is -14,33
Since the number in question is not negative the only solution is 33. Let's check our answer.

(33)(34)-20(34)=442
1,122-680=442